A radio tower is 200 feet tall. How far is an observer from the base of the tower if the angle of their line of sight from the ground to the top of the tower is 65°65°? Round to the nearest hundredth.(1 point)
Responses

    23.66 ft.
   
    428.90 ft. 
 
    178.57 ft.
    93.26 ft.

To solve this problem, we will use trigonometry. The scenario can be modeled with a right triangle where:

- The height of the tower is the opposite side (200 feet).
- The distance from the observer to the base of the tower is the adjacent side (which we need to find).
- The angle of elevation from the observer to the top of the tower is 65°.

We can use the tangent function, which relates the opposite and adjacent sides of a right triangle:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]

Substitute the known values into the formula:

\[
\tan(65^\circ) = \frac{200}{\text{adjacent}}
\]

Now, solve for the adjacent side (distance from the observer):

\[
\text{adjacent} = \frac{200}{\tan(65^\circ)}
\]

Using a calculator:

\[
\text{adjacent} \approx \frac{200}{2.1445} \approx 93.26 \text{ feet}
\]

So, the correct answer is **93.26 feet**.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do you calculate the height of a building given the angle of elevation and the distance?
2. What other trigonometric functions can you use for problems involving right triangles?
3. How does the tangent function relate to the slope of a line?
4. How would this calculation change if the angle of elevation were 45° instead of 65°?
5. How can you use trigonometry to solve real-world problems involving angles of elevation and depression?

**Tip**: When solving these problems, always check that your calculator is set to the correct mode (degrees, in this case).

A radio tower is 200 feet tall. How far is an observer from the base of the tower if the angle of their line of sight from the ground to the top of the tower is 65°? Round to the nearest hundredth.

Learn how to calculate the distance from an observer to the base of a 200-foot radio tower using trigonometry. The problem involves the use of the tangent function with an angle of elevation of 65°. The solution shows that the distance is approximately 93.26 feet, rounded to the nearest hundredth. Suitable for students in Grades 9-12.

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